<h2>Problem 265</h2>
<div style="color:#666;font-size:80%;">21 November 2009</div><br />
<div class="problem_content">
<p>2<img src="" style="display:none;" alt="^(" /><sup>N</sup><img src="" style="display:none;" alt=")" /> binary digits can be placed in a circle so that all the N-digit clockwise subsequences are distinct.</p>

<p>For N=3, two such circular arrangements are possible, ignoring rotations:</p>
<div align='center'><img src="project/images/p_265_BinaryCircles.gif" /></div>

<p>For the first arrangement, the 3-digit subsequences, in clockwise order, are:<br /> 000, 001, 010, 101, 011, 111, 110 and 100.</p>

<p>Each circular arrangement can be encoded as a number by concatenating the binary digits starting with the subsequence of all zeros as the most significant bits and proceeding clockwise. The two arrangements for N=3 are thus represented as 23 and 29:
<div align='center'>00010111&thinsp;<img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" /> = 23</div>
<div align='center'>00011101&thinsp;<img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" /> = 29</div></p>

<p>Calling S(N) the sum of the unique numeric representations, we can see that S(3) = 23 + 29 = 52.</p>

<p>Find S(5).</p>

</div><br />
